How may I figure out the scalar equation of a plane perpendicular to the plane, for example x+2y-z+3=0?

If you need points to explain, they include the origin, and the point (2,-3,2).

the normal vector of the given plane is a vector in the plane we want to find. do you see this? then, we can find another vector in the plane, by finding the vector that connects the origin to the point (2,-3,2). now that we have two vectors in our plane, we can take their cross-product. this will give us the normal vector of our plane. then we just plug everything into the formula we have for a plane, and we're done